Authors: Khovanov, Mikhail
Lauda, Aaron
Mackaay, Marco
Stošić, Marko 
Title: Extended graphical calculus for categorified quantum sl(2)
Journal: Memoirs of the American Mathematical Society
Volume: 219
Issue: 1029
First page: 1
Last page: 87
Issue Date: 1-Sep-2012
Rank: M21a
ISBN: 978-0-8218-8977-0
ISSN: 0065-9266
DOI: 10.1090/S0065-9266-2012-00665-4
A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements. These formulas have integral coefficients and imply that one of the main results of Lauda's paper- identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)-also holds when the 2-category is defined over the ring of integers rather than over a field. A new diagrammatic description of Schur functions is also given and it is shown that the the Jacobi-Trudy formulas for the decomposition of Schur functions into elementary or complete symmetric functions follows from the diagrammatic relations for categorified quantum sl(2).
Publisher: American Mathematical Society

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